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This question directly refers to the paper "Capital Asset Pricing Mistakes: The Consistent Opportunities in Tail Hedged Equities", http://www.universa.net/Universa_SpitznagelResearch_201501.pdf.

Very briefly put, an index is hedged by simultaneously buying out of the money put options (whose strike price is lower than the price of the index), so that if the index price crashes, the loss is offset by the payoff from the options. I can't exactly understand how the hedging portfolio is constructed. This is detailed in the last paragraph on page 2, which I quote here:

...we can safely ascertain that from a risk-reward standpoint, an investment in the S&P 500 Index plus short-term Treasuries could be considered a benchmark for validating a tail hedge argument. Thus, we choose a vanilla 60/40 portfolio -- 60% invested in the S&P 500 and 40% in short-term Treasuries, rebalanced monthly. On the other hand, our tail-hedged portfolio consists of S&P 500 and out-of-the-money put options (specifically one delta which has a strike roughly 30-35% below spot) on the S&P 500. At the beginning of every calendar month, using actual option prices, the number of third-month options (with a maturity from 11 to 12 weeks, and also carrying over the payoff from unexpired options) is determined such that the tail-hedged portfolio breaks even for a down 20% move in the S&P 500 over a month...

So from what I understand, suppose at the beginning of June the index price is $2000$ and a $20\%$ OTM option (Strike price of $1600$) expiring in August end costs, say, $50$. I decide to buy $x$ shares of the index and $y$ options. If the index moves down $20\%$, a potential loss of $400x+50y$ (the cost of buying options is counted as a "loss") must be offset by a gain of $400x+50y$ due to options at June end.

Does this mean that the price of each option should now be updated to $\frac{400x+50y}{y}$? Although this does make sense because the options would become at the money in case of a $20\%$ down move, obviously we can't be sure that this indeed would be the price of each option at the end of June. So do we simulate the price of each option after one month and accordingly decide how many options to buy at the start of the month (say, using Monte Carlo simulation)?

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So, you simulate the pnl one month in advance in a scenario where the Index has moved down by 20%. This is for options which are 30% + out of the money. In your example this would be August expiration and 1400 strike not the 1600 strike.

So if you are long X index shares, as you said then you would lose 400x in one month's time. You buy Y puts to hedge that. You also know how much the puts currently cost (mkt price) say P1, and from that you can figure out the implied volatility they are trading at. You have to guesstimate what the value of the puts would be in the after 1 month scenario call that P2. This basic calculator here will help: http://www.option-price.com/

The days til expiration input will change (to 8weeks), the underlying price will change (from 2000 to 1600) and Crucially the volatility will also change. It is difficult to estimate what it would be in this scenario. Typically you would expect a 20% down move in an index in 1 month to correspond to a sizeable move up in volatility. If you read up a little more on vol that might help you get a handle on how to estimate this.
https://en.wikipedia.org/wiki/Volatility_smile You can probably assume the interest rates won't change (although a tanking market might mean a rate cut) So, using your vol forecasting crystal ball you will get a price P2

Your hedged portfolio should have a pnl of 0. So Y*(P2-P1)-400X =0 You already know all of the other variables so you can solve for Y to see how many puts you should buy.

In reality the hedge will only perform as well as your estimate of the vol.

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    $\begingroup$ Yes. They probably looked at past historical episodes of "down 20% in a month" to get a feel for how much the vol will increase. $\endgroup$ – Alex C Jun 11 '16 at 16:00
  • $\begingroup$ Thanks a lot! I have a couple of questions: the basic option price calculator probably uses the Black-Scholes formula. Is it really okay to use the B-S formula to arrive at the option price? I've read that the B-S formula wrongly estimates OTM option prices and moreover, during a market crash, several Black-Scholes model assumptions may be violated. $\endgroup$ – u23 Jun 11 '16 at 17:45
  • $\begingroup$ @AlexC: Secondly, if I'm looking to implement this hedging strategy now, the method you described should work. I can just look at all the 20% down movement events from 2006 onwards and see how much the volatility went up, and that should give me a guesstimate of a potential volatility spike if the market goes down 20%. But what if I'm backtesting this? Suppose I backtest this strat from 2004 till 2012. Then for each year (say 2004) would I look at the previous 10 years data (1994-2003) and take the average of volatility spikes, for all -20% events during 1994-2003, as a guesstimate? $\endgroup$ – u23 Jun 11 '16 at 17:53
  • $\begingroup$ Yes, that's a good point. For the most objective backtest you should use data from prior episodes of volatility increase. $\endgroup$ – Alex C Jun 11 '16 at 18:15
  • $\begingroup$ EDIT to previous comment: At each point in the backtest period (say 2006), we'd look at how volatility changed with price in the years preceding it (say 1997-2006), and then estimate a volatility spike in case of a 20% price drop using some time-series estimation technique (like GARCH for example). $\endgroup$ – u23 Jun 25 '16 at 15:40

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